## The Resistance Cube, the C-Cube and the RLC-Cube

To simplify given circuits is an art of electrical engineers. Often it turns out that a complicated problem seen from a different point of view is extremely easy. The resistance cube ("R-cube") is one such example. The solution we will find can also be used to calculate the impedances of more complicated elements like the "C-cube" or the oscillatory "RLC-cube".

## Problem Statement

Find the total resistance $$R_{c}$$ of the resistance cube ("R-cube") shown below with respect to opposing edges. You may want to draw a two-dimensional circuit diagram first. There is a lot of redundancy in the cube since all resistances are equal to $$R$$ - maybe you can find a way to simplify the circuit.

When you have successfully calculated the resistance of the “R-cube”, you can easily calculate the impedance $$Z_{c,C}$$ of the “C-cube”, where all resistances are exchanged with equal capacitances $$C$$. Assume some time-harmonic voltage source connected to the opposing edges. Can you also find the impedance $$Z_{c,RLC}$$ of the “RLC-cube” as shown below with resistance $$R$$, inducance $$L$$ and capacitance $$C$$?